Loogle!

Result

Found 359 declarations mentioning Ideal.IsMaximal. Of these, only the first 200 are shown.

Ideal.IsMaximal 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] (I : Ideal α) : Prop
Ideal.exists_maximal 📋 Mathlib.RingTheory.Ideal.Maximal
(α : Type u) [Semiring α] [Nontrivial α] : ∃ M, M.IsMaximal
Ideal.IsMaximal.isPrime 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [CommSemiring α] {I : Ideal α} (H : I.IsMaximal) : I.IsPrime
Ideal.IsMaximal.isPrime' 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [CommSemiring α] (I : Ideal α) [_H : I.IsMaximal] : I.IsPrime
Ideal.IsMaximal.ne_top 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {I : Ideal α} (h : I.IsMaximal) : I ≠ ⊤
Ideal.bot_isMaximal 📋 Mathlib.RingTheory.Ideal.Maximal
{K : Type u} [DivisionSemiring K] : ⊥.IsMaximal
Ideal.IsMaximal.mk 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {I : Ideal α} (out : IsCoatom I) : I.IsMaximal
Ideal.IsMaximal.out 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} {inst✝ : Semiring α} {I : Ideal α} [self : I.IsMaximal] : IsCoatom I
Ideal.isMaximal_def 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {I : Ideal α} : I.IsMaximal ↔ IsCoatom I
Ideal.IsMaximal.lt_top 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {I : Ideal α} (h : I.IsMaximal) : I < ⊤
Ideal.irreducible_of_isMaximal_span_singleton 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [CommSemiring α] [IsDomain α] {a : α} (ha : a ≠ 0) (max : (Ideal.span {a}).IsMaximal) : Irreducible a
Ideal.exists_le_maximal 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] (I : Ideal α) (hI : I ≠ ⊤) : ∃ M, M.IsMaximal ∧ I ≤ M
Ideal.ne_top_iff_exists_maximal 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {I : Ideal α} : I ≠ ⊤ ↔ ∃ M, M.IsMaximal ∧ I ≤ M
Ideal.IsMaximal.eq_of_le 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {I J : Ideal α} (hI : I.IsMaximal) (hJ : J ≠ ⊤) (IJ : I ≤ J) : I = J
Ideal.IsMaximal.eq_iff_le 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {I J : Ideal α} (hI : I.IsMaximal) (hJ : J ≠ ⊤) : I = J ↔ I ≤ J
Ideal.irreducible_of_isMaximal_of_eq_span_singleton_of_not_isIdempotentElem 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [CommSemiring α] {a : α} (max : (Ideal.span {a}).IsMaximal) (idem : ∀ (x : α), Ideal.span {a} = Ideal.span {x} → ¬IsIdempotentElem x) : Irreducible a
Ideal.IsMaximal.coprime_of_ne 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {M M' : Ideal α} (hM : M.IsMaximal) (hM' : M'.IsMaximal) (hne : M ≠ M') : M ⊔ M' = ⊤
Ideal.maximal_of_no_maximal 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {P : Ideal α} (hmax : ∀ (m : Ideal α), P < m → ¬m.IsMaximal) (J : Ideal α) (hPJ : P < J) : J = ⊤
Ideal.IsMaximal.exists_inv 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {I : Ideal α} (hI : I.IsMaximal) {x : α} (hx : x ∉ I) : ∃ y, ∃ i ∈ I, y * x + i = 1
Ideal.isMaximal_iff 📋 Mathlib.RingTheory.Ideal.Maximal
{α : Type u} [Semiring α] {I : Ideal α} : I.IsMaximal ↔ 1 ∉ I ∧ ∀ (J : Ideal α) (x : α), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J
Ring.isField_iff_maximal_bot 📋 Mathlib.RingTheory.Ideal.Basic
{R : Type u_4} [CommSemiring R] [Nontrivial R] : IsField R ↔ ⊥.IsMaximal
Ring.ne_bot_of_isMaximal_of_not_isField 📋 Mathlib.RingTheory.Ideal.Basic
{R : Type u_4} [CommSemiring R] [Nontrivial R] {M : Ideal R} (max : M.IsMaximal) (not_field : ¬IsField R) : M ≠ ⊥
Ring.exists_maximal_of_not_isField 📋 Mathlib.RingTheory.Ideal.Basic
{R : Type u_4} [CommSemiring R] [Nontrivial R] (h : ¬IsField R) : ∃ p, p ≠ ⊥ ∧ p.IsMaximal
Ideal.bot_lt_of_maximal 📋 Mathlib.RingTheory.Ideal.Basic
{R : Type u_4} [CommSemiring R] [Nontrivial R] (M : Ideal R) [hm : M.IsMaximal] (non_field : ¬IsField R) : ⊥ < M
Ideal.Quotient.divisionRing 📋 Mathlib.RingTheory.Ideal.Quotient.Basic
{R : Type u_3} [Ring R] (I : Ideal R) [I.IsTwoSided] [I.IsMaximal] : DivisionRing (R ⧸ I)
Ideal.Quotient.field 📋 Mathlib.RingTheory.Ideal.Quotient.Basic
{R : Type u_4} [CommRing R] (I : Ideal R) [I.IsMaximal] : Field (R ⧸ I)
Ideal.Quotient.groupWithZero 📋 Mathlib.RingTheory.Ideal.Quotient.Basic
{R : Type u_3} [Ring R] (I : Ideal R) [I.IsTwoSided] [hI : I.IsMaximal] : GroupWithZero (R ⧸ I)
Ideal.Quotient.maximal_of_isField 📋 Mathlib.RingTheory.Ideal.Quotient.Basic
{R : Type u_4} [CommRing R] (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal
Ideal.Quotient.maximal_ideal_iff_isField_quotient 📋 Mathlib.RingTheory.Ideal.Quotient.Basic
{R : Type u_4} [CommRing R] (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I)
Ideal.Quotient.exists_inv 📋 Mathlib.RingTheory.Ideal.Quotient.Basic
{R : Type u_3} [Ring R] {I : Ideal R} [I.IsTwoSided] [hI : I.IsMaximal] {a : R ⧸ I} : a ≠ 0 → ∃ b, a * b = 1
Ideal.isCoprime_of_isMaximal 📋 Mathlib.RingTheory.Ideal.Operations
{R : Type u} [CommSemiring R] {I J : Ideal R} [I.IsMaximal] [J.IsMaximal] (ne : I ≠ J) : IsCoprime I J
Ideal.IsMaximal.exists_inv_pow 📋 Mathlib.RingTheory.Ideal.Operations
{R : Type u} [CommSemiring R] (I : Ideal R) [I.IsMaximal] {x : R} (hx : x ∉ I) (n : ℕ) : ∃ y, ∃ i ∈ I ^ n, y * x + i = 1
Ideal.IsMaximal.mem_pow_mul 📋 Mathlib.RingTheory.Ideal.Operations
{R : Type u_2} [CommSemiring R] (I : Ideal R) [I.IsMaximal] {a b : R} {n : ℕ} (h : a * b ∈ I ^ n) : a ∈ I ^ n ∨ b ∈ I
Ideal.IsMaximal.mul_mem_pow 📋 Mathlib.RingTheory.Ideal.Operations
{R : Type u} [CommSemiring R] (I : Ideal R) [I.IsMaximal] {a b : R} {n : ℕ} (h : a * b ∈ I ^ n) : a ∈ I ∨ b ∈ I ^ n
Ideal.subset_iUnion_iff_mem_of_isMaximal_of_finite 📋 Mathlib.RingTheory.Ideal.Operations
{R : Type u_2} [CommRing R] {M : Ideal R} [M.IsMaximal] {S : Set (Ideal R)} (hs : S.Finite) (a b : Ideal R) (hp : ∀ I ∈ S, I ≠ a → I ≠ b → I.IsPrime) (ha : a ≠ ⊤) (hb : b ≠ ⊤) : ↑M ⊆ ⋃ I ∈ S, ↑I ↔ M ∈ S
Ideal.IsMaximal.map_bijective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Bijective ⇑f) {I : Ideal R} : I.IsMaximal → (Ideal.map f I).IsMaximal
Ideal.isMaximal_map_iff_of_bijective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Bijective ⇑f) {I : Ideal R} : (Ideal.map f I).IsMaximal ↔ I.IsMaximal
Ideal.IsMaximal.comap_bijective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Bijective ⇑f) {K : Ideal S} : K.IsMaximal → (Ideal.comap f K).IsMaximal
Ideal.isMaximal_comap_iff_of_bijective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Bijective ⇑f) {K : Ideal S} : (Ideal.comap f K).IsMaximal ↔ K.IsMaximal
RingHom.ker_isMaximal_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u_1} {K : Type u_2} {F : Type u_3} [Ring R] [DivisionRing K] [FunLike F R K] [RingHomClass F R K] (f : F) (hf : Function.Surjective ⇑f) : (RingHom.ker f).IsMaximal
Ideal.map_isMaximal_of_equiv 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] {E : Type u_4} [EquivLike E R S] [RingEquivClass E R S] (e : E) {p : Ideal R} [hp : p.IsMaximal] : (Ideal.map e p).IsMaximal
Ideal.comap_isMaximal_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) {K : Ideal S} [H : K.IsMaximal] : (Ideal.comap f K).IsMaximal
Ideal.isMaximal_iff_of_bijective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Bijective ⇑f) : ⊥.IsMaximal ↔ ⊥.IsMaximal
Ideal.comap_isMaximal_of_equiv 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] {E : Type u_4} [EquivLike E R S] [RingEquivClass E R S] (e : E) {p : Ideal S} [hp : p.IsMaximal] : (Ideal.comap e p).IsMaximal
Ideal.map_eq_top_or_isMaximal_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Surjective ⇑f) {I : Ideal R} (H : I.IsMaximal) : Ideal.map f I = ⊤ ∨ (Ideal.map f I).IsMaximal
Ideal.IsMaximal.comap_piEvalRingHom 📋 Mathlib.RingTheory.Ideal.Maps
{ι : Type u_4} {R : ι → Type u_5} [(i : ι) → Semiring (R i)] {i : ι} {I : Ideal (R i)} (h : I.IsMaximal) : (Ideal.comap (Pi.evalRingHom R i) I).IsMaximal
Ideal.IsMaximal.map_of_surjective_of_ker_le 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u_1} {S : Type u_2} {F : Type u_3} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} (hf : Function.Surjective ⇑f) {m : Ideal R} [m.IsMaximal] (hk : RingHom.ker f ≤ m) : (Ideal.map f m).IsMaximal
Ideal.comap_map_eq_self_of_isMaximal 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) {p : Ideal R} [hP' : p.IsMaximal] (hP : Ideal.map f p ≠ ⊤) : Ideal.comap f (Ideal.map f p) = p
exists_max_ideal_of_mem_nonunits 📋 Mathlib.RingTheory.Ideal.Nonunits
{α : Type u_2} {a : α} [CommSemiring α] (h : a ∈ nonunits α) : ∃ I, I.IsMaximal ∧ a ∈ I
PrincipalIdealRing.isMaximal_of_irreducible 📋 Mathlib.RingTheory.PrincipalIdealDomain
{R : Type u} [CommSemiring R] [IsPrincipalIdealRing R] {p : R} (hp : Irreducible p) : Ideal.IsMaximal (R ∙ p)
Ideal.irreducible_iff_isMaximal_span_singleton_of_not_isField 📋 Mathlib.RingTheory.PrincipalIdealDomain
{R : Type u} [CommSemiring R] [IsPrincipalIdealRing R] [IsDomain R] (h : ¬IsField R) {p : R} : Irreducible p ↔ Ideal.IsMaximal (R ∙ p)
Ideal.irreducible_iff_isMaximal_span_singleton 📋 Mathlib.RingTheory.PrincipalIdealDomain
{R : Type u} [CommSemiring R] [IsPrincipalIdealRing R] [IsDomain R] {p : R} (hp : p ≠ 0) : Irreducible p ↔ Ideal.IsMaximal (R ∙ p)
IsPrime.to_maximal_ideal 📋 Mathlib.RingTheory.PrincipalIdealDomain
{R : Type u} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] {S : Ideal R} [hpi : S.IsPrime] (hS : S ≠ ⊥) : S.IsMaximal
Ideal.eq_zero_of_constant_mem_of_maximal 📋 Mathlib.RingTheory.Polynomial.Basic
{R : Type u} [Ring R] (hR : IsField R) (I : Ideal (Polynomial R)) [hI : I.IsMaximal] (x : R) (hx : Polynomial.C x ∈ I) : x = 0
Ideal.bot_quotient_isMaximal_iff 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R : Type u} [Ring R] (I : Ideal R) [I.IsTwoSided] : ⊥.IsMaximal ↔ I.IsMaximal
Ring.jacobson_eq_sInf_isMaximal 📋 Mathlib.RingTheory.Jacobson.Radical
(R : Type u_1) [Ring R] : Ring.jacobson R = sInf {I | I.IsMaximal}
Ring.jacobson_le_of_isMaximal 📋 Mathlib.RingTheory.Jacobson.Radical
{R : Type u_1} [Ring R] (m : Ideal R) [m.IsMaximal] : Ring.jacobson R ≤ m
Ideal.jacobson.isMaximal 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} [Ring R] {I : Ideal R} [H : I.IsMaximal] : I.jacobson.IsMaximal
Ideal.jacobson_eq_self_of_isMaximal 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} [Ring R] {I : Ideal R} [H : I.IsMaximal] : I.jacobson = I
Ideal.isLocal_of_isMaximal_radical 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} [CommRing R] {I : Ideal R} (hi : I.radical.IsMaximal) : I.IsLocal
Ideal.IsLocal.mk 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} [CommRing R] {I : Ideal R} (out : I.jacobson.IsMaximal) : I.IsLocal
Ideal.IsLocal.out 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} {inst✝ : CommRing R} {I : Ideal R} [self : I.IsLocal] : I.jacobson.IsMaximal
Ideal.isLocal_iff 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} [CommRing R] {I : Ideal R} : I.IsLocal ↔ I.jacobson.IsMaximal
Ideal.eq_jacobson_iff_sInf_maximal 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} [Ring R] {I : Ideal R} : I.jacobson = I ↔ ∃ M, (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M
Ideal.eq_jacobson_iff_notMem 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} [Ring R] {I : Ideal R} : I.jacobson = I ↔ ∀ x ∉ I, ∃ M, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
Ideal.comap_jacobson 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} {K : Ideal S} : Ideal.comap f K.jacobson = sInf (Ideal.comap f '' {J | K ≤ J ∧ J.IsMaximal})
IsLocalRing.of_unique_max_ideal 📋 Mathlib.RingTheory.LocalRing.Basic
{R : Type u_1} [CommSemiring R] (h : ∃! I, I.IsMaximal) : IsLocalRing R
MaximalSpectrum.isMaximal 📋 Mathlib.RingTheory.Spectrum.Maximal.Defs
{R : Type u_1} [CommSemiring R] (self : MaximalSpectrum R) : self.asIdeal.IsMaximal
MaximalSpectrum.mk 📋 Mathlib.RingTheory.Spectrum.Maximal.Defs
{R : Type u_1} [CommSemiring R] (asIdeal : Ideal R) (isMaximal : asIdeal.IsMaximal) : MaximalSpectrum R
IsLocalRing.maximalIdeal.isMaximal 📋 Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
(R : Type u_1) [CommSemiring R] [IsLocalRing R] : (IsLocalRing.maximalIdeal R).IsMaximal
IsLocalRing.maximal_ideal_unique 📋 Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
(R : Type u_1) [CommSemiring R] [IsLocalRing R] : ∃! I, I.IsMaximal
IsLocalRing.eq_maximalIdeal 📋 Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
{R : Type u_1} [CommSemiring R] [IsLocalRing R] {I : Ideal R} (hI : I.IsMaximal) : I = IsLocalRing.maximalIdeal R
IsLocalRing.isMaximal_iff 📋 Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
(R : Type u_1) [CommSemiring R] [IsLocalRing R] {I : Ideal R} : I.IsMaximal ↔ I = IsLocalRing.maximalIdeal R
Ideal.under_map_eq_map_under 📋 Mathlib.RingTheory.Ideal.Over
{A : Type u_2} [CommSemiring A] {B : Type u_3} [CommSemiring B] [Algebra A B] (P : Ideal B) {C : Type u_5} {D : Type u_6} [CommSemiring C] [Semiring D] [Algebra A C] [Algebra C D] [Algebra A D] [Algebra B D] [IsScalarTower A C D] [IsScalarTower A B D] (h₁ : (Ideal.map (algebraMap A C) (Ideal.under A P)).IsMaximal) (h₂ : Ideal.map (algebraMap B D) P ≠ ⊤) : Ideal.under C (Ideal.map (algebraMap B D) P) = Ideal.map (algebraMap A C) (Ideal.under A P)
Ideal.isPrimary_of_isMaximal_radical 📋 Mathlib.RingTheory.Ideal.IsPrimary
{R : Type u_1} [CommSemiring R] {I : Ideal R} (hi : I.radical.IsMaximal) : I.IsPrimary
IsLocalization.isMaximal_of_isMaximal_under 📋 Mathlib.RingTheory.Localization.Ideal
{R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (I : Ideal S) [hI : (Ideal.under R I).IsMaximal] : I.IsMaximal
IsLocalization.isMaximal_of_isMaximal_disjoint 📋 Mathlib.RingTheory.Localization.Ideal
{R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (I : Ideal R) [hI : I.IsMaximal] (h : Disjoint ↑M ↑I) : (Ideal.map (algebraMap R S) I).IsMaximal
IsLocalization.surjective_quotientMap_of_maximal_of_localization 📋 Mathlib.RingTheory.Localization.Ideal
{R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] {I : Ideal S} [I.IsPrime] {J : Ideal R} {H : J ≤ Ideal.under R I} (hI : (Ideal.under R I).IsMaximal) : Function.Surjective ⇑(Ideal.quotientMap I (algebraMap R S) H)
Ideal.Quotient.isUnit_mk_pow_of_notMem 📋 Mathlib.RingTheory.Ideal.Quotient.Nilpotent
{S : Type u_1} [CommRing S] (I : Ideal S) [I.IsMaximal] {n : ℕ} {x : S} (hx : x ∉ I) : IsUnit ((Ideal.Quotient.mk (I ^ n)) x)
Ideal.Quotient.isUnit_mk_pow_iff_notMem 📋 Mathlib.RingTheory.Ideal.Quotient.Nilpotent
{S : Type u_1} [CommRing S] (I : Ideal S) [I.IsMaximal] {n : ℕ} (hn : n ≠ 0) {x : S} : IsUnit ((Ideal.Quotient.mk (I ^ n)) x) ↔ x ∉ I
Ideal.IsMaximal.of_isLocalization_of_disjoint 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] [IsLocalization M S] (I : Ideal S) [hI : (Ideal.under R I).IsMaximal] : I.IsMaximal
IsLocalization.AtPrime.isMaximal_map 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] (p : Ideal R) [p.IsPrime] (Rₚ : Type u_4) [CommSemiring Rₚ] [Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [IsLocalRing Rₚ] : (Ideal.map (algebraMap R Rₚ) p).IsMaximal
IsLocalization.AtPrime.comap_map_of_isMaximal 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (p : Ideal R) [p.IsPrime] (Sₚ : Type u_5) [CommSemiring Sₚ] [Algebra S Sₚ] [IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] (P : Ideal S) [P.IsMaximal] [P.LiesOver p] : Ideal.under S (Ideal.map (algebraMap S Sₚ) P) = P
IsLocalization.AtPrime.under_map_of_isMaximal 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] (S : Type u_2) [CommSemiring S] [Algebra R S] (p : Ideal R) [p.IsPrime] (Sₚ : Type u_5) [CommSemiring Sₚ] [Algebra S Sₚ] [IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] (P : Ideal S) [P.IsMaximal] [P.LiesOver p] : Ideal.under S (Ideal.map (algebraMap S Sₚ) P) = P
IsLocalization.AtPrime.comap_maximalIdeal_pow 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] (p : Ideal R) [p.IsPrime] (Rₚ : Type u_4) [CommSemiring Rₚ] [Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [IsLocalRing Rₚ] [p.IsMaximal] (n : ℕ) : Ideal.under R (IsLocalRing.maximalIdeal Rₚ ^ n) = p ^ n
IsLocalization.AtPrime.under_maximalIdeal_pow 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] (p : Ideal R) [p.IsPrime] (Rₚ : Type u_4) [CommSemiring Rₚ] [Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [IsLocalRing Rₚ] [p.IsMaximal] (n : ℕ) : Ideal.under R (IsLocalRing.maximalIdeal Rₚ ^ n) = p ^ n
IsLocalization.AtPrime.equivQuotMaximalIdeal 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_7} [CommRing R] (p : Ideal R) [p.IsMaximal] (Rₚ : Type u_8) [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [IsLocalRing Rₚ] : R ⧸ p ≃+* Rₚ ⧸ IsLocalRing.maximalIdeal Rₚ
IsLocalization.AtPrime.comap_map_eq_map 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{S : Type u_6} {R : Type u_7} [CommRing R] (p : Ideal R) [p.IsMaximal] {Sₚ : Type u_9} [CommRing S] [Algebra R S] [CommRing Sₚ] [Algebra S Sₚ] [Algebra R Sₚ] [IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] [IsScalarTower R S Sₚ] : Ideal.under S (Ideal.map (algebraMap R Sₚ) p) = Ideal.map (algebraMap R S) p
IsLocalization.AtPrime.under_map_eq_map 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{S : Type u_6} {R : Type u_7} [CommRing R] (p : Ideal R) [p.IsMaximal] {Sₚ : Type u_9} [CommRing S] [Algebra R S] [CommRing Sₚ] [Algebra S Sₚ] [Algebra R Sₚ] [IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] [IsScalarTower R S Sₚ] : Ideal.under S (Ideal.map (algebraMap R Sₚ) p) = Ideal.map (algebraMap R S) p
IsLocalization.AtPrime.equivQuotMaximalIdealPow 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_7} [CommRing R] (p : Ideal R) [p.IsMaximal] (Rₚ : Type u_8) [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [IsLocalRing Rₚ] (n : ℕ) : (R ⧸ p ^ n) ≃ₐ[R] Rₚ ⧸ IsLocalRing.maximalIdeal Rₚ ^ n
IsLocalization.AtPrime.equivQuotMaximalIdeal_apply_mk 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_7} [CommRing R] (p : Ideal R) [p.IsMaximal] (Rₚ : Type u_8) [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [IsLocalRing Rₚ] (x : R) : (IsLocalization.AtPrime.equivQuotMaximalIdeal p Rₚ) ((Ideal.Quotient.mk p) x) = (Ideal.Quotient.mk (IsLocalRing.maximalIdeal Rₚ)) ((algebraMap R Rₚ) x)
IsLocalization.AtPrime.equivQuotientMapMaximalIdeal 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
(S : Type u_6) {R : Type u_7} [CommRing R] (p : Ideal R) [p.IsMaximal] (Rₚ : Type u_8) [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [IsLocalRing Rₚ] (Sₚ : Type u_9) [CommRing S] [Algebra R S] [CommRing Sₚ] [Algebra S Sₚ] [Algebra R Sₚ] [Algebra Rₚ Sₚ] [IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] [IsScalarTower R S Sₚ] [IsScalarTower R Rₚ Sₚ] : S ⧸ Ideal.map (algebraMap R S) p ≃+* Sₚ ⧸ Ideal.map (algebraMap Rₚ Sₚ) (IsLocalRing.maximalIdeal Rₚ)
IsLocalization.AtPrime.equivQuotMaximalIdeal_symm_apply_mk 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_7} [CommRing R] (p : Ideal R) [p.IsMaximal] (Rₚ : Type u_8) [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [IsLocalRing Rₚ] (x : R) (s : ↥p.primeCompl) : (IsLocalization.AtPrime.equivQuotMaximalIdeal p Rₚ).symm ((Ideal.Quotient.mk (IsLocalRing.maximalIdeal Rₚ)) (IsLocalization.mk' Rₚ x s)) = (Ideal.Quotient.mk p) x * ((Ideal.Quotient.mk p) ↑s)⁻¹
IsLocalization.AtPrime.equivQuotMaximalIdealPow_apply_mk 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_7} [CommRing R] (p : Ideal R) [p.IsMaximal] (Rₚ : Type u_8) [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [IsLocalRing Rₚ] (n : ℕ) (x : R) : (IsLocalization.AtPrime.equivQuotMaximalIdealPow p Rₚ n) ((Ideal.Quotient.mk (p ^ n)) x) = (Ideal.Quotient.mk (IsLocalRing.maximalIdeal Rₚ ^ n)) ((algebraMap R Rₚ) x)
IsLocalization.AtPrime.equivQuotMaximalIdealPow_symm_apply_mk_mul 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_7} [CommRing R] (p : Ideal R) [p.IsMaximal] (Rₚ : Type u_8) [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [IsLocalRing Rₚ] (n : ℕ) (x : R) (s : ↥p.primeCompl) : (IsLocalization.AtPrime.equivQuotMaximalIdealPow p Rₚ n).symm ((Ideal.Quotient.mk (IsLocalRing.maximalIdeal Rₚ ^ n)) (IsLocalization.mk' Rₚ x s)) * (Ideal.Quotient.mk (p ^ n)) ↑s = (Ideal.Quotient.mk (p ^ n)) x
Module.subsingleton_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] (h : ∀ (P : Ideal R) [inst : P.IsMaximal], Subsingleton (Mₚ P)) : Subsingleton M
Submodule.eq_of_localization₀_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] {N₁ N₂ : Submodule R M} (h : ∀ (P : Ideal R) [inst : P.IsMaximal], Submodule.localized₀ P.primeCompl (f P) N₁ = Submodule.localized₀ P.primeCompl (f P) N₂) : N₁ = N₂
Submodule.eq_bot_of_localization₀_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] (N : Submodule R M) (h : ∀ (P : Ideal R) [inst : P.IsMaximal], Submodule.localized₀ P.primeCompl (f P) N = ⊥) : N = ⊥
Submodule.eq_top_of_localization₀_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] (N : Submodule R M) (h : ∀ (P : Ideal R) [inst : P.IsMaximal], Submodule.localized₀ P.primeCompl (f P) N = ⊤) : N = ⊤
Module.eq_zero_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] (m : M) (h : ∀ (P : Ideal R) [inst : P.IsMaximal], (f P) m = 0) : m = 0
Module.eq_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] (m m' : M) (h : ∀ (P : Ideal R) [inst : P.IsMaximal], (f P) m = (f P) m') : m = m'
Submodule.le_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] {N₁ N₂ : Submodule R M} (h : ∀ (P : Ideal R) [inst : P.IsMaximal], Submodule.localized₀ P.primeCompl (f P) N₁ ≤ Submodule.localized₀ P.primeCompl (f P) N₂) : N₁ ≤ N₂
Submodule.mem_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] (m : M) (N : Submodule R M) (h : ∀ (P : Ideal R) [inst : P.IsMaximal], (f P) m ∈ Submodule.localized₀ P.primeCompl (f P) N) : m ∈ N
Submodule.eq_bot_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Rₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4) [(P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)] [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalization.AtPrime (Rₚ P) P] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module (Rₚ P) (Mₚ P)] [∀ (P : Ideal R) [inst : P.IsMaximal], IsScalarTower R (Rₚ P) (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] (N : Submodule R M) (h : ∀ (P : Ideal R) [inst : P.IsMaximal], Submodule.localized' (Rₚ P) P.primeCompl (f P) N = ⊥) : N = ⊥
Submodule.eq_top_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Rₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4) [(P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)] [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalization.AtPrime (Rₚ P) P] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module (Rₚ P) (Mₚ P)] [∀ (P : Ideal R) [inst : P.IsMaximal], IsScalarTower R (Rₚ P) (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] (N : Submodule R M) (h : ∀ (P : Ideal R) [inst : P.IsMaximal], Submodule.localized' (Rₚ P) P.primeCompl (f P) N = ⊤) : N = ⊤
Submodule.eq_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Rₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4) [(P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)] [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalization.AtPrime (Rₚ P) P] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module (Rₚ P) (Mₚ P)] [∀ (P : Ideal R) [inst : P.IsMaximal], IsScalarTower R (Rₚ P) (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] {N₁ N₂ : Submodule R M} (h : ∀ (P : Ideal R) [inst : P.IsMaximal], Submodule.localized' (Rₚ P) P.primeCompl (f P) N₁ = Submodule.localized' (Rₚ P) P.primeCompl (f P) N₂) : N₁ = N₂
LinearMap.eq_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Submodule
{R : Type u_1} {M : Type u_2} {M₁ : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₁] [Module R M₁] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)] (M₁ₚ : (P : Ideal R) → [P.IsMaximal] → Type u_6) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (M₁ₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (M₁ₚ P)] (f₁ : (P : Ideal R) → [inst : P.IsMaximal] → M₁ →ₗ[R] M₁ₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f₁ P)] (g g' : M →ₗ[R] M₁) (h : ∀ (P : Ideal R) [inst : P.IsMaximal], (IsLocalizedModule.map P.primeCompl (f P) (f₁ P)) g = (IsLocalizedModule.map P.primeCompl (f P) (f₁ P)) g') : g = g'
LinearIndependent.of_isLocalized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (Rₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5) [(P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)] [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalization.AtPrime (Rₚ P) P] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_6) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module (Rₚ P) (Mₚ P)] [∀ (P : Ideal R) [inst : P.IsMaximal], IsScalarTower R (Rₚ P) (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (f P)] {ι : Type u_9} (v : ι → M) (H : ∀ (P : Ideal R) [inst : P.IsMaximal], LinearIndependent (Rₚ P) (⇑(f P) ∘ v)) : LinearIndependent R v
bijective_of_isLocalization_isMaximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] [Algebra R S] (Rₚ : (p : Ideal R) → [p.IsMaximal] → Type u_3) [(p : Ideal R) → [inst : p.IsMaximal] → CommSemiring (Rₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra R (Rₚ p)] (Sₚ : (p : Ideal R) → [p.IsMaximal] → Type u_4) [(p : Ideal R) → [inst : p.IsMaximal] → CommSemiring (Sₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra S (Sₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra (Rₚ p) (Sₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra R (Sₚ p)] [∀ (p : Ideal R) [inst : p.IsMaximal], IsScalarTower R (Rₚ p) (Sₚ p)] [∀ (p : Ideal R) [inst : p.IsMaximal], IsScalarTower R S (Sₚ p)] [∀ (p : Ideal R) [inst : p.IsMaximal], IsLocalization.AtPrime (Rₚ p) p] [∀ (p : Ideal R) [inst : p.IsMaximal], IsLocalizedModule.AtPrime p ↑(IsScalarTower.toAlgHom R S (Sₚ p))] (H : ∀ (p : Ideal R) [inst : p.IsMaximal], Function.Bijective ⇑(algebraMap (Rₚ p) (Sₚ p))) : Function.Bijective ⇑(algebraMap R S)
injective_of_isLocalization_isMaximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] [Algebra R S] (Rₚ : (p : Ideal R) → [p.IsMaximal] → Type u_3) [(p : Ideal R) → [inst : p.IsMaximal] → CommSemiring (Rₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra R (Rₚ p)] (Sₚ : (p : Ideal R) → [p.IsMaximal] → Type u_4) [(p : Ideal R) → [inst : p.IsMaximal] → CommSemiring (Sₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra S (Sₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra (Rₚ p) (Sₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra R (Sₚ p)] [∀ (p : Ideal R) [inst : p.IsMaximal], IsScalarTower R (Rₚ p) (Sₚ p)] [∀ (p : Ideal R) [inst : p.IsMaximal], IsScalarTower R S (Sₚ p)] [∀ (p : Ideal R) [inst : p.IsMaximal], IsLocalization.AtPrime (Rₚ p) p] [∀ (p : Ideal R) [inst : p.IsMaximal], IsLocalizedModule.AtPrime p ↑(IsScalarTower.toAlgHom R S (Sₚ p))] (H : ∀ (p : Ideal R) [inst : p.IsMaximal], Function.Injective ⇑(algebraMap (Rₚ p) (Sₚ p))) : Function.Injective ⇑(algebraMap R S)
surjective_of_isLocalization_isMaximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] [Algebra R S] (Rₚ : (p : Ideal R) → [p.IsMaximal] → Type u_3) [(p : Ideal R) → [inst : p.IsMaximal] → CommSemiring (Rₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra R (Rₚ p)] (Sₚ : (p : Ideal R) → [p.IsMaximal] → Type u_4) [(p : Ideal R) → [inst : p.IsMaximal] → CommSemiring (Sₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra S (Sₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra (Rₚ p) (Sₚ p)] [(p : Ideal R) → [inst : p.IsMaximal] → Algebra R (Sₚ p)] [∀ (p : Ideal R) [inst : p.IsMaximal], IsScalarTower R (Rₚ p) (Sₚ p)] [∀ (p : Ideal R) [inst : p.IsMaximal], IsScalarTower R S (Sₚ p)] [∀ (p : Ideal R) [inst : p.IsMaximal], IsLocalization.AtPrime (Rₚ p) p] [∀ (p : Ideal R) [inst : p.IsMaximal], IsLocalizedModule.AtPrime p ↑(IsScalarTower.toAlgHom R S (Sₚ p))] (H : ∀ (p : Ideal R) [inst : p.IsMaximal], Function.Surjective ⇑(algebraMap (Rₚ p) (Sₚ p))) : Function.Surjective ⇑(algebraMap R S)
bijective_of_isLocalized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_6) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (f P)] (Nₚ : (P : Ideal R) → [P.IsMaximal] → Type u_7) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Nₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Nₚ P)] (g : (P : Ideal R) → [inst : P.IsMaximal] → N →ₗ[R] Nₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (g P)] (F : M →ₗ[R] N) (H : ∀ (P : Ideal R) [inst : P.IsMaximal], Function.Bijective ⇑((IsLocalizedModule.map P.primeCompl (f P) (g P)) F)) : Function.Bijective ⇑F
injective_of_isLocalized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_6) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (f P)] (Nₚ : (P : Ideal R) → [P.IsMaximal] → Type u_7) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Nₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Nₚ P)] (g : (P : Ideal R) → [inst : P.IsMaximal] → N →ₗ[R] Nₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (g P)] (F : M →ₗ[R] N) (H : ∀ (P : Ideal R) [inst : P.IsMaximal], Function.Injective ⇑((IsLocalizedModule.map P.primeCompl (f P) (g P)) F)) : Function.Injective ⇑F
surjective_of_isLocalized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_6) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (f P)] (Nₚ : (P : Ideal R) → [P.IsMaximal] → Type u_7) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Nₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Nₚ P)] (g : (P : Ideal R) → [inst : P.IsMaximal] → N →ₗ[R] Nₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (g P)] (F : M →ₗ[R] N) (H : ∀ (P : Ideal R) [inst : P.IsMaximal], Function.Surjective ⇑((IsLocalizedModule.map P.primeCompl (f P) (g P)) F)) : Function.Surjective ⇑F
exact_of_isLocalized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {L : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid L] [Module R L] (Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_6) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)] (f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (f P)] (Nₚ : (P : Ideal R) → [P.IsMaximal] → Type u_7) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Nₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Nₚ P)] (g : (P : Ideal R) → [inst : P.IsMaximal] → N →ₗ[R] Nₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (g P)] (Lₚ : (P : Ideal R) → [P.IsMaximal] → Type u_8) [(P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Lₚ P)] [(P : Ideal R) → [inst : P.IsMaximal] → Module R (Lₚ P)] (h : (P : Ideal R) → [inst : P.IsMaximal] → L →ₗ[R] Lₚ P) [∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (h P)] (F : M →ₗ[R] N) (G : N →ₗ[R] L) (H : ∀ (J : Ideal R) [inst : J.IsMaximal], Function.Exact ⇑((IsLocalizedModule.map J.primeCompl (f J) (g J)) F) ⇑((IsLocalizedModule.map J.primeCompl (g J) (h J)) G)) : Function.Exact ⇑F ⇑G
bijective_of_localized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (f : M →ₗ[R] N) (h : ∀ (J : Ideal R) [inst : J.IsMaximal], Function.Bijective ⇑((LocalizedModule.map J.primeCompl) f)) : Function.Bijective ⇑f
injective_of_localized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (f : M →ₗ[R] N) (h : ∀ (J : Ideal R) [inst : J.IsMaximal], Function.Injective ⇑((LocalizedModule.map J.primeCompl) f)) : Function.Injective ⇑f
surjective_of_localized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (f : M →ₗ[R] N) (h : ∀ (J : Ideal R) [inst : J.IsMaximal], Function.Surjective ⇑((LocalizedModule.map J.primeCompl) f)) : Function.Surjective ⇑f
exact_of_localized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {L : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid L] [Module R L] (f : M →ₗ[R] N) (g : N →ₗ[R] L) (h : ∀ (J : Ideal R) [inst : J.IsMaximal], Function.Exact ⇑((LocalizedModule.map J.primeCompl) f) ⇑((LocalizedModule.map J.primeCompl) g)) : Function.Exact ⇑f ⇑g
Module.flat_of_localized_maximal 📋 Mathlib.RingTheory.Flat.Localization
{R : Type u_1} [CommSemiring R] (M : Type u_3) [AddCommMonoid M] [Module R M] (h : ∀ (P : Ideal R) [inst : P.IsMaximal], Module.Flat R (LocalizedModule P.primeCompl M)) : Module.Flat R M
Module.flat_of_isLocalized_maximal 📋 Mathlib.RingTheory.Flat.Localization
{R : Type u_1} (S : Type u_2) [CommSemiring R] [CommSemiring S] [Algebra R S] (M : Type u_3) [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] (Mₚ : (P : Ideal S) → [P.IsMaximal] → Type u_4) [(P : Ideal S) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)] [(P : Ideal S) → [inst : P.IsMaximal] → Module R (Mₚ P)] [(P : Ideal S) → [inst : P.IsMaximal] → Module S (Mₚ P)] [∀ (P : Ideal S) [inst : P.IsMaximal], IsScalarTower R S (Mₚ P)] (f : (P : Ideal S) → [inst : P.IsMaximal] → M →ₗ[S] Mₚ P) [∀ (P : Ideal S) [inst : P.IsMaximal], IsLocalizedModule.AtPrime P (f P)] (H : ∀ (P : Ideal S) [inst : P.IsMaximal], Module.Flat R (Mₚ P)) : Module.Flat R M
eq_zero_of_localization 📋 Mathlib.RingTheory.LocalProperties.Basic
{R : Type u_1} [CommSemiring R] (r : R) (h : ∀ (J : Ideal R) (x : J.IsMaximal), (algebraMap R (Localization.AtPrime J)) r = 0) : r = 0
ideal_eq_bot_of_localization' 📋 Mathlib.RingTheory.LocalProperties.Basic
{R : Type u_1} [CommSemiring R] (I : Ideal R) (h : ∀ (J : Ideal R) (x : J.IsMaximal), Ideal.map (algebraMap R (Localization.AtPrime J)) I = ⊥) : I = ⊥
Ideal.eq_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Basic
{R : Type u_1} [CommSemiring R] {I J : Ideal R} (h : ∀ (P : Ideal R) (x : P.IsMaximal), Ideal.map (algebraMap R (Localization.AtPrime P)) I = Ideal.map (algebraMap R (Localization.AtPrime P)) J) : I = J
ideal_eq_bot_of_localization 📋 Mathlib.RingTheory.LocalProperties.Basic
{R : Type u_1} [CommSemiring R] (I : Ideal R) (h : ∀ (J : Ideal R) (x : J.IsMaximal), IsLocalization.coeSubmodule (Localization.AtPrime J) I = ⊥) : I = ⊥
Ideal.mem_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Basic
{R : Type u_1} [CommSemiring R] {r : R} {J : Ideal R} (h : ∀ (P : Ideal R) (x : P.IsMaximal), (algebraMap R (Localization.AtPrime P)) r ∈ Ideal.map (algebraMap R (Localization.AtPrime P)) J) : r ∈ J
Ideal.le_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Basic
{R : Type u_1} [CommSemiring R] {I J : Ideal R} (h : ∀ (P : Ideal R) (x : P.IsMaximal), Ideal.map (algebraMap R (Localization.AtPrime P)) I ≤ Ideal.map (algebraMap R (Localization.AtPrime P)) J) : I ≤ J
Ideal.IsMaximal.ne_bot_of_isIntegral_int 📋 Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
{R : Type u_1} [CommRing R] [CharZero R] [Algebra.IsIntegral ℤ R] (I : Ideal R) [I.IsMaximal] : I ≠ ⊥
Algebra.ker_algebraMap_isMaximal_of_isIntegral 📋 Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
(R : Type u_1) [CommRing R] (k : Type u_6) [Field k] [Algebra R k] [Algebra.IsIntegral R k] : (RingHom.ker (algebraMap R k)).IsMaximal
Ideal.IsMaximal.under 📋 Mathlib.RingTheory.Ideal.GoingUp
(A : Type u_1) [CommRing A] {B : Type u_2} [CommRing B] [Algebra A B] [Algebra.IsIntegral A B] (P : Ideal B) [P.IsMaximal] : (Ideal.under A P).IsMaximal
Ideal.IsMaximal.of_isMaximal_liesOver 📋 Mathlib.RingTheory.Ideal.GoingUp
{A : Type u_1} [CommRing A] {B : Type u_2} [CommRing B] [Algebra A B] [Algebra.IsIntegral A B] (P : Ideal B) (p : Ideal A) [P.LiesOver p] [P.IsMaximal] : p.IsMaximal
Ideal.IsMaximal.of_liesOver_isMaximal 📋 Mathlib.RingTheory.Ideal.GoingUp
{A : Type u_1} [CommRing A] {B : Type u_2} [CommRing B] [Algebra A B] [Algebra.IsIntegral A B] (P : Ideal B) (p : Ideal A) [P.LiesOver p] [hpm : p.IsMaximal] [P.IsPrime] : P.IsMaximal
Ideal.exists_maximal_ideal_liesOver_of_isIntegral 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] [FaithfulSMul R S] (P : Ideal R) [P.IsMaximal] : ∃ Q, Q.IsMaximal ∧ Q.LiesOver P
Ideal.isMaximal_of_mem_primesOver 📋 Mathlib.RingTheory.Ideal.GoingUp
{A : Type u_1} [CommRing A] {p : Ideal A} [p.IsMaximal] {B : Type u_2} [CommRing B] [Algebra A B] [Algebra.IsIntegral A B] {P : Ideal B} (hP : P ∈ p.primesOver B) : P.IsMaximal
Ideal.isMaximal_comap_of_isIntegral_of_isMaximal' 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_3} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) (hf : f.IsIntegral) (I : Ideal S) [I.IsMaximal] : (Ideal.comap f I).IsMaximal
Ideal.primesOver.isMaximal 📋 Mathlib.RingTheory.Ideal.GoingUp
{A : Type u_1} [CommRing A] {p : Ideal A} [p.IsMaximal] {B : Type u_2} [CommRing B] [Algebra A B] [Algebra.IsIntegral A B] (Q : ↑(p.primesOver B)) : (↑Q).IsMaximal
Ideal.isMaximal_comap_of_isIntegral_of_isMaximal 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (I : Ideal S) [hI : I.IsMaximal] : (Ideal.comap (algebraMap R S) I).IsMaximal
Ideal.isMaximal_of_isIntegral_of_isMaximal_comap' 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] (f : R →+* S) (hf : f.IsIntegral) (I : Ideal S) [I.IsPrime] (hI : (Ideal.comap f I).IsMaximal) : I.IsMaximal
Ideal.isMaximal_of_isIntegral_of_isMaximal_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (I : Ideal S) [I.IsPrime] (hI : (Ideal.comap (algebraMap R S) I).IsMaximal) : I.IsMaximal
Ideal.IntegralClosure.isMaximal_of_isMaximal_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] (I : Ideal A) [I.IsPrime] (hI : (Ideal.comap (algebraMap R A) I).IsMaximal) : I.IsMaximal
Ideal.IsIntegral.isMaximal_of_isMaximal_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] (I : Ideal A) [I.IsPrime] (hI : (Ideal.comap (algebraMap R A) I).IsMaximal) : I.IsMaximal
Ideal.IsIntegralClosure.isMaximal_of_isMaximal_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] (I : Ideal A) [I.IsPrime] (hI : (Ideal.comap (algebraMap R A) I).IsMaximal) : I.IsMaximal
Ideal.exists_ideal_over_maximal_of_isIntegral 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (P : Ideal R) [P_max : P.IsMaximal] (hP : RingHom.ker (algebraMap R S) ≤ P) : ∃ Q, Q.IsMaximal ∧ Ideal.comap (algebraMap R S) Q = P
Module.FaithfullyFlat.submodule_ne_top 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
{R : Type u} {M : Type v} {inst✝ : CommRing R} {inst✝¹ : AddCommGroup M} {inst✝² : Module R M} [self : Module.FaithfullyFlat R M] ⦃m : Ideal R⦄ : m.IsMaximal → m • ⊤ ≠ ⊤
Module.FaithfullyFlat.mk 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
{R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] [toFlat : Module.Flat R M] (submodule_ne_top : ∀ ⦃m : Ideal R⦄, m.IsMaximal → m • ⊤ ≠ ⊤) : Module.FaithfullyFlat R M
Module.faithfullyFlat_iff 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
(R : Type u) (M : Type v) [CommRing R] [AddCommGroup M] [Module R M] : Module.FaithfullyFlat R M ↔ Module.Flat R M ∧ ∀ ⦃m : Ideal R⦄, m.IsMaximal → m • ⊤ ≠ ⊤
PrimeSpectrum.isMax_iff 📋 Mathlib.RingTheory.Spectrum.Prime.Basic
{R : Type u} [CommSemiring R] {x : PrimeSpectrum R} : IsMax x ↔ x.asIdeal.IsMaximal
PrimeSpectrum.zeroLocus_eq_singleton 📋 Mathlib.RingTheory.Spectrum.Prime.Basic
{R : Type u} [CommSemiring R] (m : Ideal R) [m.IsMaximal] : PrimeSpectrum.zeroLocus ↑m = {{ asIdeal := m, isPrime := ⋯ }}
RingHom.surjectiveOnStalks_iff_forall_maximal' 📋 Mathlib.RingTheory.SurjectiveOnStalks
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} : f.SurjectiveOnStalks ↔ ∀ (I : Ideal S), I.IsMaximal → ∀ (s : S), ∃ x r, ∃ c ∉ I, f r ∉ I ∧ c * f r * s = c * f x
RingHom.surjectiveOnStalks_iff_forall_maximal 📋 Mathlib.RingTheory.SurjectiveOnStalks
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} : f.SurjectiveOnStalks ↔ ∀ (I : Ideal S) (x : I.IsMaximal), Function.Surjective ⇑(Localization.localRingHom (Ideal.comap f I) I f ⋯)
instFiniteResidueField 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{R : Type u_1} [CommRing R] (I : Ideal R) [I.IsMaximal] : Module.Finite R I.ResidueField
Ideal.bijective_algebraMap_quotient_residueField 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{R : Type u_1} [CommRing R] (I : Ideal R) [I.IsMaximal] : Function.Bijective ⇑(algebraMap (R ⧸ I) I.ResidueField)
Ideal.algebraMap_residueField_surjective 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{R : Type u_1} [CommRing R] (I : Ideal R) [I.IsMaximal] : Function.Surjective ⇑(algebraMap R I.ResidueField)
PrimeSpectrum.exists_maximal_notMem_range_sigmaToPi_of_infinite 📋 Mathlib.RingTheory.Spectrum.Prime.RingHom
{ι : Type u_3} (R : ι → Type u_2) [(i : ι) → CommSemiring (R i)] [Infinite ι] [∀ (i : ι), Nontrivial (R i)] : ∃ I, ∃ (x : I.IsMaximal), { asIdeal := I, isPrime := ⋯ } ∉ Set.range (PrimeSpectrum.sigmaToPi R)
instIsMaximalOfIsPrimeOfKrullDimLEOfNatNat 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] (I : Ideal R) [I.IsPrime] [Ring.KrullDimLE 0 R] : I.IsMaximal
Ideal.isMaximal_of_isPrime 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] [Ring.KrullDimLE 0 R] (I : Ideal R) [I.IsPrime] : I.IsMaximal
Ideal.IsPrime.isMaximal' 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] [Ring.KrullDimLE 0 R] {I : Ideal R} (hI : I.IsPrime) : I.IsMaximal
Ring.KrullDimLE.mk₀ 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] (H : ∀ (I : Ideal R), I.IsPrime → I.IsMaximal) : Ring.KrullDimLE 0 R
Ideal.isMaximal_iff_isPrime 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] [Ring.KrullDimLE 0 R] {I : Ideal R} : I.IsMaximal ↔ I.IsPrime
Ring.krullDimLE_zero_iff 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] : Ring.KrullDimLE 0 R ↔ ∀ (I : Ideal R), I.IsPrime → I.IsMaximal
Ring.krullDimLE_zero_iff_forall_minimalPrimes_isMaximal 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] : Ring.KrullDimLE 0 R ↔ ∀ I ∈ minimalPrimes R, I.IsMaximal
Prime.isMaximal_span_singleton 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] [NoZeroDivisors R] [Ring.KrullDimLE 1 R] {a : R} (ha : Prime a) : (Ideal.span {a}).IsMaximal
Ring.KrullDimLE.mk₁ 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] (H : ∀ (I : Ideal R), I.IsPrime → I ∈ minimalPrimes R ∨ I.IsMaximal) : Ring.KrullDimLE 1 R
Ring.krullDimLE_one_iff 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] : Ring.KrullDimLE 1 R ↔ ∀ (I : Ideal R), I.IsPrime → I ∈ minimalPrimes R ∨ I.IsMaximal
Ring.KrullDimLE.mk₁' 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] (H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → I.IsMaximal) : Ring.KrullDimLE 1 R
Ideal.isMaximal_of_isPrime_of_ne_bot 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] [NoZeroDivisors R] [Ring.KrullDimLE 1 R] (I : Ideal R) [I.IsPrime] (hI' : I ≠ ⊥) : I.IsMaximal
Ideal.IsPrime.isMaximal_of_ne_bot 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] [NoZeroDivisors R] [Ring.KrullDimLE 1 R] {I : Ideal R} (hI : I.IsPrime) (hI' : I ≠ ⊥) : I.IsMaximal
Ring.krullDimLE_one_iff_of_noZeroDivisors 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] [NoZeroDivisors R] : Ring.KrullDimLE 1 R ↔ ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → I.IsMaximal
Ring.krullDimLE_one_iff_of_isPrime_bot 📋 Mathlib.RingTheory.KrullDimension.Basic
{R : Type u_1} [CommSemiring R] [⊥.IsPrime] : Ring.KrullDimLE 1 R ↔ ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → I.IsMaximal
MaximalSpectrum.equivSubtype 📋 Mathlib.RingTheory.Spectrum.Maximal.Basic
(R : Type u_1) [CommSemiring R] : MaximalSpectrum R ≃ { I // I.IsMaximal }
MaximalSpectrum.range_asIdeal 📋 Mathlib.RingTheory.Spectrum.Maximal.Basic
(R : Type u_1) [CommSemiring R] : Set.range MaximalSpectrum.asIdeal = {J | J.IsMaximal}
MaximalSpectrum.equivSubtype_apply_coe 📋 Mathlib.RingTheory.Spectrum.Maximal.Basic
(R : Type u_1) [CommSemiring R] (I : MaximalSpectrum R) : ↑((MaximalSpectrum.equivSubtype R) I) = I.asIdeal
MaximalSpectrum.equivSubtype_symm_apply_asIdeal 📋 Mathlib.RingTheory.Spectrum.Maximal.Basic
(R : Type u_1) [CommSemiring R] (I : { I // I.IsMaximal }) : ((MaximalSpectrum.equivSubtype R).symm I).asIdeal = ↑I
PrimeSpectrum.piLocalizationToMaximalEquiv 📋 Mathlib.RingTheory.Spectrum.Maximal.Localization
{R : Type u_1} [CommSemiring R] (h : ∀ (I : Ideal R), I.IsPrime → I.IsMaximal) : PrimeSpectrum.PiLocalization R ≃+* MaximalSpectrum.PiLocalization R
PrimeSpectrum.isMaximal_of_toPiLocalization_surjective 📋 Mathlib.RingTheory.Spectrum.Maximal.Localization
{R : Type u_1} [CommSemiring R] (surj : Function.Surjective ⇑(PrimeSpectrum.toPiLocalization R)) (I : PrimeSpectrum R) : I.asIdeal.IsMaximal
PrimeSpectrum.piLocalizationToMaximal_bijective 📋 Mathlib.RingTheory.Spectrum.Maximal.Localization
{R : Type u_1} [CommSemiring R] (h : ∀ (I : Ideal R), I.IsPrime → I.IsMaximal) : Function.Bijective ⇑(PrimeSpectrum.piLocalizationToMaximal R)
PrimeSpectrum.isClosed_singleton_iff_isMaximal 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
{R : Type u} [CommSemiring R] (x : PrimeSpectrum R) : IsClosed {x} ↔ x.asIdeal.IsMaximal
PrimeSpectrum.stableUnderSpecialization_singleton 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
{R : Type u} [CommSemiring R] {x : PrimeSpectrum R} : StableUnderSpecialization {x} ↔ x.asIdeal.IsMaximal
PrimeSpectrum.discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
{R : Type u} [CommSemiring R] : DiscreteTopology (PrimeSpectrum R) ↔ Finite ↑{I | I.IsMaximal} ∧ sInf {I | I.IsMaximal} ≤ nilradical R
IsSimpleModule.annihilator_isMaximal 📋 Mathlib.RingTheory.SimpleModule.Basic
{M : Type u_4} [AddCommGroup M] {R : Type u_6} [CommRing R] [Module R M] [simple : IsSimpleModule R M] : (Module.annihilator R M).IsMaximal
IsSimpleModule.ker_toSpanSingleton_isMaximal 📋 Mathlib.RingTheory.SimpleModule.Basic
(R : Type u_2) [Ring R] {M : Type u_4} [AddCommGroup M] [Module R M] [IsSimpleModule R M] {m : M} (hm : m ≠ 0) : Ideal.IsMaximal (LinearMap.toSpanSingleton R M m).ker
isSimpleModule_iff_quot_maximal 📋 Mathlib.RingTheory.SimpleModule.Basic
{R : Type u_2} [Ring R] {M : Type u_4} [AddCommGroup M] [Module R M] : IsSimpleModule R M ↔ ∃ I, I.IsMaximal ∧ Nonempty (M ≃ₗ[R] R ⧸ I)
MixedCharZero.reduce_to_maximal_ideal 📋 Mathlib.Algebra.CharP.MixedCharZero
(R : Type u_1) [CommRing R] {p : ℕ} (hp : Nat.Prime p) : (∃ I, I ≠ ⊤ ∧ CharP (R ⧸ I) p) ↔ ∃ I, I.IsMaximal ∧ CharP (R ⧸ I) p
IsArtinianRing.setOfPred_isMaximal_finite 📋 Mathlib.RingTheory.Artinian.Module
(R : Type u_1) [CommSemiring R] [IsArtinianRing R] : {I | I.IsMaximal}.Finite
IsArtinianRing.setOf_isMaximal_finite 📋 Mathlib.RingTheory.Artinian.Module
(R : Type u_1) [CommSemiring R] [IsArtinianRing R] : {I | I.IsMaximal}.Finite
IsArtinianRing.isMaximal_of_isPrime 📋 Mathlib.RingTheory.Artinian.Module
{R : Type u_2} [CommRing R] (p : Ideal R) [p.IsPrime] [IsArtinianRing R] : p.IsMaximal
IsArtinianRing.isPrime_iff_isMaximal 📋 Mathlib.RingTheory.Artinian.Module
{R : Type u_1} [CommRing R] [IsArtinianRing R] (p : Ideal R) : p.IsPrime ↔ p.IsMaximal
AdjoinRoot.span_maximal_of_irreducible 📋 Mathlib.RingTheory.AdjoinRoot
{K : Type u_5} [Field K] {f : Polynomial K} [Fact (Irreducible f)] : (Ideal.span {f}).IsMaximal
AlgebraicClosure.maxIdeal.isMaximal 📋 Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
(k : Type u) [Field k] : (AlgebraicClosure.maxIdeal k).IsMaximal
Ideal.IsPrime.isMaximal 📋 Mathlib.RingTheory.DedekindDomain.Basic
{R : Type u_4} [CommRing R] [Ring.DimensionLEOne R] {p : Ideal R} (h : p.IsPrime) (hp : p ≠ ⊥) : p.IsMaximal
Ring.DimensionLEOne.maximalOfPrime 📋 Mathlib.RingTheory.DedekindDomain.Basic
{R : Type u_1} {inst✝ : CommRing R} [self : Ring.DimensionLEOne R] {p : Ideal R} : p ≠ ⊥ → p.IsPrime → p.IsMaximal
Ring.DimensionLEOne.mk 📋 Mathlib.RingTheory.DedekindDomain.Basic
{R : Type u_1} [CommRing R] (maximalOfPrime : ∀ {p : Ideal R}, p ≠ ⊥ → p.IsPrime → p.IsMaximal) : Ring.DimensionLEOne R
IsLocalRing.primesOver_eq 📋 Mathlib.RingTheory.DedekindDomain.Basic
{R : Type u_1} (A : Type u_2) [CommRing R] [CommRing A] [IsLocalRing A] [IsDedekindDomain A] [Algebra R A] [FaithfulSMul R A] [Module.Finite R A] {p : Ideal R} [p.IsMaximal] (hp0 : p ≠ ⊥) : p.primesOver A = {IsLocalRing.maximalIdeal A}
Ideal.jacobson_bot_polynomial_le_sInf_map_maximal 📋 Mathlib.RingTheory.Jacobson.Polynomial
{R : Type u_1} [CommRing R] : ⊥.jacobson ≤ sInf (Ideal.map Polynomial.C '' {J | J.IsMaximal})
IsLocalization.isMaximal_iff_isMaximal_disjoint 📋 Mathlib.RingTheory.Jacobson.Ring
{R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [H : IsJacobsonRing R] (J : Ideal S) : J.IsMaximal ↔ (Ideal.under R J).IsMaximal ∧ y ∉ Ideal.under R J
IsLocalization.isMaximal_of_isMaximal_notMem 📋 Mathlib.RingTheory.Jacobson.Ring
{R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] (I : Ideal R) (hI : I.IsMaximal) (hy : y ∉ I) : (Ideal.map (algebraMap R S) I).IsMaximal
isJacobsonRing_iff_sInf_maximal 📋 Mathlib.RingTheory.Jacobson.Ring
{R : Type u_1} [CommRing R] : IsJacobsonRing R ↔ ∀ {I : Ideal R}, I.IsPrime → ∃ M, (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M
Polynomial.isMaximal_comap_C_of_isJacobsonRing 📋 Mathlib.RingTheory.Jacobson.Ring
{R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal] [IsJacobsonRing R] : (Ideal.comap Polynomial.C P).IsMaximal
IsLocalization.orderIsoOfMaximal 📋 Mathlib.RingTheory.Jacobson.Ring
{R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [IsJacobsonRing R] : { p // p.IsMaximal } ≃o { p // p.IsMaximal ∧ y ∉ p }
Polynomial.quotient_mk_comp_C_isIntegral_of_isJacobsonRing 📋 Mathlib.RingTheory.Jacobson.Ring
{R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal] [IsJacobsonRing R] : ((Ideal.Quotient.mk P).comp Polynomial.C).IsIntegral

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c